-( Story related, cannot be missed )-
This is the second part of the Lumpy Ice Princess case. As such, you will normally get this, but if you still want some help, refer to the spoiler tags.
*** Spoiler - click to reveal ***Act 2: Lumpy Ice Princess
Part 2a: The Guest Quest
Ice King is a rather unruly house guest. After all of his demands, he'll be satisfied with just one: getting Gunter to come over. Head downstairs and converse with BMO to try and put on the disguise of being Gunter. It'll be excited to do everything but won't agree to a playdate with Ice King in its foul mood. Head back upstairs, then climb further up the same ladder to get to Jake and Finn's bedroom and collect the tape on the far left. Return and give the tape to BMO to get its mood up and ready to dress up. Now, we have to collect some things scattered throughout the house: white paint can be found in the crow's nest (third floor, the right ladder leading up); black paint can be found outside (first floor, outside the treasury); paintbrushes can be found in the play room (third floor, far right from Ice King); glue in the kitchen (left of BMO on the second floor); some Meat Man Meatballs from the freezer (in the kitchen, second floor); and a gold foil card in the armory (ladder leading down, between BMO and the kitchen in the second floor).
Part 2b: Three Noble Deeds
After talking with LSP, Pillowmint Butler, and Princess Bubblegum, claim the shield in the foyer then head out into the hall. Talk to Doctor Donut, then head behind him into the kitchen. Here, we want the spatula and the invitation. After that, head further down the hall and collect some cake from the wall they're repairing as well as the spray can before it in front of the Security Room. Then you're free to leave the palace. You can go ahead and give the note to Nurse Poundcake to solve the first deed. Head down the stairs to find the snail for this part, then talk to the Marshmallow Kid, Mr. Creampuff, and Cinnamon Bun. You can also collect one of the candy rockets.
Head down the street between Mr. Creampuff and Cinnamon to encounter the newest resident of the Ooo, Sunny, as well as a feather hanging out in a trash can for you to pick up. Also here you can talk to Cookie Guy to find that he'll trade you his rope belt for something sugary. Return to the festivities and use the spray can to extract some punch. Take this back to Cookie Guy to get the rope, then combine it with the rocket to get a fuse. Run further right and use it on fire, then quickly run to the right some more and use it on the licorice steel lock. Then you can enter and find Dr. J with Jake 2. Luckily, you already have something for him to fetch in the form of the handlebars. All that's left is to put together the cake for the Marshmallow Kid. Use the shield to the right of him on the other table, then place the cake on it. For frosting, use the feather on Mr. Creampuff, then top it off with the sparkling gem you got from the forest earlier.